The statistics of Pearce element diagrams and the Chayes closure problem

Pearce element ratios are defined as having a constituent in their denominator that is conserved in a system undergoing change. The presence of a conserved element in the denominator simplifies the statistics of such ratios and renders them subject to statistical tests, especially tests of significance of the correlation coefficient between Pearce element ratios. Pearce element ratio diagrams provide unambigous tests of petrologic hypotheses because they are based on the stoichiometry of rock-forming minerals. There are three ways to recognize a conserved element: 1. The petrologic behavior of the element can be used to select conserved ones. They are usually the incompatible elements. 2. The ratio of two conserved elements will be constant in a comagmatic suite. 3. An element ratio diagram that is not constructed with a conserved element in the denominator will have a trend with a near zero intercept. The last two criteria can be tested statistically. The significance of the slope, intercept and correlation coefficient can be tested by estimating the probability of obtaining the observed values from a random population of arrays. This population of arrays must satisfy two criteria: 1. The population must contain at least one array that has the means and variances of the array of analytical data for the rock suite. 2. Arrays with the means and variances of the data must not be so abundant in the population that nearly every array selected at random has the properties of the data. The population of random closed arrays can be obtained from a population of open arrays whose elements are randomly selected from probability distributions. The means and variances of these probability distributions are themselves selected from probability distributions which have means and variances equal to a hypothetical open array that would give the means and variances of the data on closure. This hypothetical open array is called the Chayes array. Alternatively, the population of random closed arrays can be drawn from the compositional space available to rock-forming processes. The minerals comprising the available space can be described with one additive component per mineral phase and a small number of exchange components. This space is called Thompson space. Statistics based on either space lead to the conclusion that Pearce element ratios are statistically valid and that Pearce element diagrams depict the processes that create chemical inhomogeneities in igneous rock suites.     

The validity of pearce element ratio analysis in petrology: an example from the Uwekahuna laccolith,Hawaii

Chemical variation diagrams formed from ratios with a common denominator exhibit induced correlations which can obscure the effects of true compositional variations. Nevertheless, important conclusions have been obtained using Pearce element ratio (and isochron) diagrams, even though these diagrams are formed from axes ratios with a common denominator. In this paper, we consider two approaches to mitigate the effects of induced correlation, and in doing so, demonstrate the validity of geochemical analysis using diagrams formed from ratios with a common denominator. First, we propagate analytical error onto Pearce element ratio diagrams to distinguish chemical variations associated with analytical error from chemical variations related to geological processes. Second, we consider an alternative diagram formed from ratios with different denominators which does not exhibit any induced correlation. We demonstrate both theoretically and by example that results produced using this diagram are identical to those using diagrams with a common denominator. These two lines of evidence confirm that Pearce element ratio (and isochron) diagrams are useful and valid tools in the analysis of geochemical variations.     

The adaptation of Pearce element ratio diagrams to complex high silica systems

Pearce element ratios (PERs, of Pearce 1968) express geochemical data in a form where variations in absolute compositions of an igneous suite can be evaluated. Generally the denominator value in the ratio is taken as a major element abundance, but it is argued here that Zr provides a more suitable choice. Zr remains incompatible in magmatic systems up to 68 wt.% SiO₂ because zircon fractionation can be suppressed by high melt temperatures and increased volatile contents. The use of Zr thus permits PER modelling to be extended to much higher levels of silica than previously investigated. However, such systems are more complex than those just involving simple basaltic magmas. Besides fractionation, the processes of magma mixing, combined assimilation and fractional crystallization, and the initial degree of partial melting in the mantle source must also be considered. To distinguish and evaluate these processes a set of example suites are investigated from a complex synextensional calc-alkaline province in the western USA. Samples within most individual suites can be modelled by fractionation, however a significant trend orthogonal to the main fractionation vector is also apparent, and open system processes are inferred. Successful modelling is achieved on an inter-suite basis using diagrams with axis functions of ([4(Ca+Na)+0.5(Fe+Mg)]/Zr versus (Si+Al)/Zr). Potential open system evolution paths between mafic end members and crustal contaminants are also displayed and evaluated on these same diagrams. The encouraging results suggest that such PER diagrams may be employed as a versatile tool for investigating the systematics of related igneous suites over a wide area.     

The dependence of partition coefficients on differences of ionic volumes in crystal-melt systems

Plots of partition coefficients versus differences of ionic volumes have been constructed from reported values in olivine-, pyroxene-, hornblende-, biotite-, and plagioclase-groundmass systems. The trend behaviour of partition coefficients as a function of the differences of ionic volumes is largely mineralogically controlled. The slopes of the trend lines can be related to the volume work resulting either from (i) expansion of the host lattice surrounding the substituting ions or (ii) collapse of the surrounding lattice if small ions are incorporated. When the volume work is in the order of the enthalpy of melting, ion selectivity ceases.If melts change from basic to highly acidic the chemical control of element partitioning by complex formation in the fluid phase is recognizable. Mineralogical versus chemical control is discussed by means of the trends of the partition coefficients of REE.The proposed ln D vs. r ³ plots improve the physical interpretation of element partition, delineates methods for interpolation or extrapolation of unknown data. Deviations of extreme data from linear relations demonstrate the constraints of our knowledge on diadochy. Rules of element partition as recently proposed by Philpotts (1978) are critically discussed.     

Artificial isochrons

The rubidium-strontium isochron procedure has proven to be a powerful tool in many geochemical studies. An isochron is said to exist if a plot of (⁸⁷Sr/⁸⁶Sr) versus (⁸⁷Rb/⁸⁶Sr) produces a linear trend with a large correlation coefficient. However, the variables selected to portray the isochron have the same denominator (⁸⁶Sr) and Pearson long ago noted that a large correlation could be induced when such ratios are formed from uncorrelated numerators and denominators. A set of numerical experiments are described that illustrate the common denominator effects as applied specifically to rubidium-strontium systematics. For at least one previously published RbSr isochron it can be shown that the common denominator effect is capable of producing a correlation coefficient that is very nearly 1.000. However, it is also shown that, for the data sets analyzed, the common denominator effect can not produce a geologically meaningful isochron. The numerical approach to assessing the common denominator effect can be applied only to those sets of isotopic analyses in which ⁸⁷Rb and ⁸⁶Sr have been determined by isotope dilution techniques. For the many data sets in which only the ratios (⁸⁷Sr/⁸⁶Sr) and (⁸⁷Rb/⁸⁶Sr) have been determined the common denominator effect can neither be assessed nor dismissed as trivial.     

Report of the International Association of Geoanalysts on the Certification of Penrhyn Slate, OU-6

The International Association of Geoanalysts (IAG) has certified a slate sample, OU-6, for twelve major and minor constituents, as well as thirty-five trace elements through an interlaboratory programme conducted in close compliance with the International Organization for Standardization (ISO) Guide 35 (1989). Laboratories were qualified for participation in the certification programme, based on their performance in the prior analysis of OU-6 as a proficiency test material. Thirty laboratories provided data for the certification, though not for each constituent that was certified. Certification criteria included a means of establishing traceability for the certification data, generally through concurrent analysis of the existing Geological Survey of Japan reference material JSl-1, agreement of results between laboratories and methods, and a minimum of data rejection (4% rejection rate for OU-6 data and 6% for JSl-1 data), preferably for well-understood technical reasons only. Information values are provided for an additional eight constituents where certification criteria were not met. Uncertainties developed in accordance with the "Guide on Uncertainty in Measurement" (Eurachem 2000) and representing the 95% confidence interval of the certified and information values are reported for all fifty five constituents. The material is currently available in 40 g units for distribution by the IAG. Supply is anticipated to last about ten years.     

Rare earth elements in alpine peridotites

The samples from alpine peridotite massifs (Beni Bouchera, Lherz and the Alps) have been analyzed for rare earth elements. The peridotites as a whole are characterized by various degrees of light REE depletion (Ce varies from 1.2 to 0.02 times chondrite) and a small variation in heavy REE (Yb varies about a factor of 2, from 1.3 to 2.2 times chondrite). They show a restricted and regular distribution in a Ce-Yb diagram, giving two types of linear trends for individual massifs (trend A for the Alps and Lizard; trend B for Beni Bouchera and Lherz, branching from trend A). The model calculations of partial melting based on the partition relations of REE among silicate minerals and melts suggest that trend A could represent a series of residua left after partial melting of garnet peridotite despite the fact that there is no garnet observed in the peridotites studied here. It is suggested that trend A would represent a melting event which predated the emplacement of the massifs and occurred at higher pressure (in the presence of garnet) than expected from the present mineralogy. The calculations also suggest that trend B could represent a partial melting event at lower pressures than trend A after the massifs uplifted into spinel peridotite field. It is also suggested that the REE concentrations of the mantle could be estimated as 2–2.5 times chondrite.     

Modelling of igneous fractionation and other processes using Pearce diagrams

Geochemical data can be quantitatively modelled by means of Pearce diagrams. These are graphs of A/Z vs B/Z where A, B and Z are compositional abundances (e.g. wt.% SiO₂, wt.% MgO, and ppm La) and Z has the additional property of having constant absolute abundance. In the terminology of igneous petrology, Z (the common denominator variable) could be an incompatible element. The numerators (A and B) may be complex algebraic combinations of elements, or even CIPW normative abundances. The utility of Pearce diagrams lies in the fact that slopes of data distributions equal the bulk AB ratio of minerals lost or gained from a suite of cogenetic rocks. There is no distortion because these plots correct for data closure. Terms of the form A_i·Z₀/Z_i (where Z₀ is the abundance in a reference sample) remove the scaling to A_i caused by the abundance of a particular choice of Z. Subtraction of these terms for different samples (e.g. A_i·(Z₀/Z_i)-A_j· (Z₀/Z_j)) quantifies mineral losses and gains. Mathematical analysis shows that limited compatibility of the denominator variable is permitted. A bulk partition value (D) of 0.1 introduces an error of only 10% in values of A_i***-Z₀/Z_i, and 10° in slope-angle on Pearce diagrams over a crystallization interval of 50%. For D0.01 the error is minimal for a crystallization interval over 90%.     

Threshold of sediment motion under unidirectional currents

Carefully selected data for the threshold of sediment movement under unidirectional flow conditions have been utilized to re-examine the various empirical curves that are commonly employed to predict this threshold. After a review of the existing data, we employed only that data obtained from open channel flumes with parallel sidewalls where flows were uniform and steady over flattened beds of unigranular, rounded sediments. Without these restrictions, an unmanageable amount of scatter is introduced. This selected data is used to develop a modified Shields-type threshold diagram that extends the limits of the original diagram by three orders of magnitude in the grain-Reynolds number. The equally general but more easily employed Yalin diagram for sediment threshold is also examined. Although the Shields and Yalin diagrams are general in that they apply to a wide range of different liquids, in both cases somewhat different curves are obtained for threshold under air than for the liquids. The often used empirical curves of the friction velocity u_*, the velocity 100 cm above the bed u₁₀₀, the bottom stress θ_t, and Shields’ relative stress θ_t, all versus the grain diameter D, are limited in their ranges of application to certain combinations of grain density, fluid density, fluid viscosity and gravity. These conditions must be selected before the curves are generated from either the more general Shields or Yalin curves. For example, on the basis of the data selected for use in this paper, empirical threshold relationships for quartz density material in water are where the velocity u₁₀₀ measured 100 cm above the sediment bed is given in cm/sec and the grain diameter D is in cm. The limitations on any of the threshold relationships are severe. These limitations should be properly understood so that the empirical curves and relationships are not improperly employed.     

The character of the fine fractions of the bottom sediments of the Bohemian part of the Labe (Elbe) river

The amounts of fractions with size 63 m in the bottom sediments of the Bohemian part of the Labe (Elbe) river is variable. The grain-size distribution is poorly to very poorly sorted with a mean particle size from 5.1 to 12.2 m.The crystalline phase of recent sediments of the Labe river draining various types of parent rocks has a simple mineral composition, with predominant mica/illite, quartz, kaolinite and chlorite (±vermiculite ±smectite). There are also very small amounts of interstratification of 10 Å and 14 Å sheet silicates (11–13.6 Å). Small amounts of Na-rich plagioclase, K-feldspar and calcite have also been found; accessory amounts of amphibole, authigenic gypsum and pyrite are also present. Opal in diatomic tests is an omnipresent constituent. In the amorphic inorganic phase, hydrated oxides predominate, where Si >Al >Fe >Mn in a ratio of 64:28:22:1. The greatest amount of amorphous inorganic and organic phases were found in the 4 m fraction.A considerable part of the detrital material originates from soils formed on various igneous, metamorphic and sedimentary rocks, transported to the Labe by other rivers or directly by erosion in the immediate vicinity of the Labe river.     

Petrologic hypothesis testing with Pearce element ratio diagrams: derivation of diagram axes

In petrology, Pearce element ratio (PER) diagrams have been used: i) to determine whether members of a rock suite are co-genetic, ii) to identify the minerals involved in differentiation processes, and iii) to evaluate the extent to which those mineral are involved. The axis coefficients of each diagram are chosen such that sorting of minerals or combinations of minerals will generate unique and predictable trends. Unfortunately, selection of the optimal combination of axis coefficients is a difficult task, especially if the system being investigated has a large number of phases or complicated solid solution minerals. Our work has established a formal set of rules and matrix operations which facilitate the determination of PER diagram axes coefficients. This methodology can be used to determine the unit molar vector displacement caused by the addition or subtraction of a specific mineral, given a set of axis coefficients. It can also be used to create PER diagrams on which minerals have predetermined vector displacements. By designating all vector displacements to be parallel, axis coefficients for assemblage test diagrams can be determined to test the following hypothesis: the observed chemical variation is due to the addition (or removal) of a specific set of minerals. Alternatively, by designating all vector displacements to be mutually perpendicular, phase discrimination diagrams can be created which test whether the observed chemical variations require a specific phase to be involved in differentiation. Phase discrimination diagrams also provide a means to estimate the extent of that involvement. This methodology facilitates construction of powerful yet simple PER diagrams which provide an effective means of testing alternative differentiation hypotheses. Current address: Department of Geological Sciences, Queen's University Kingston, Ontario K7L 3N6, Canada     

Evolution cristallochimique des amphiboles dans la série granitique de Fort-Trinquet (Mauritanie)

The precambrian postorogenic pluton of the Fort-Trinquet area (Northern Mauretania) is composed by a series of granitic rocks in which amphiboles are the characteristic mafic minerals. Twenty six amphiboles have been separated and chemically analysed; the optical properties and the unit-cell data are also given.The crystallographic and chemical differences between these minerals reflect the variations in bulk composition of the host-rocks. In the plutonic suite, two igneous trends have been recognized: 1) a granitic trend (quartz monzonite-adamellite-granite) and 2) a syenitic trend (syenite-quartz bearing syenite-alkali granite). The amphiboles of the first trend belong to the tremolite-hastingsite series; they become richer in hastingsite mole with increasing proportion of quartz and decreasing Mg/(Mg+Fe) ratio in the rocks. The granitoids of the second trend are characterized by the occurrence of two amphiboles: a primary prismatic green-coloured actinolitic hornblende generally surrounded by a dark blue rim of riebeckite composition; the riebeckite may also form some acicular crystals associated to needles of stilpnomelane. These coexisting amphiboles would result from autometasomatic reactions which affected the rocks of the syenitic trend and which gave rise to the late alkali pegmatites where the constituent is a low arfvedsonitic riebeckite. The major substitutions involved in that amphibole transformation are Na^x R³⁺Ca^x R²⁺ and Na^xSiCa^xAl^IV.Comparison with experimental data allows to estimate the physical conditions during the emplacement and the tardimagmatic evolution of this granitoid series.     

Temporal evolution and particle size distribution of aerosol constituents collected in Northern Tunisia (Boukornine) under Sirocco wind circulations

Aerosols samples were collected from Boukornine, Northern Tunisia in order to examine the temporal evolution of the principal constituents of the Tunisian aerosols under Sirocco wind circulations. High constituent concentrations registered during the period of Sirocco showed the contribution of the Sahara in terms of loading aerosols with particulate matters. The mass size distribution of the principal constituents of the aerosols under Sirocco wind circulations was obtained from measurements carried out with an inertial cascade impactor. These distributions of the aerosol showed a bimodal structure for the majority of the elements except sulfur and sodium. The mass size distribution has a significant fraction of their mass concentration in the fine particle size range. This may be explained by the effect of wind wakes and probably selective disintegration.     

Systematic use of trace elements in igneous processes

This paper develops an application of the inverse method proposed by Allégre et al. (1977) and Minster et al. (1977). Using a set of trace element data in a suite of primary lavas, it is possible to statistically test whether the data can be described by batch partial melting. Once this is achieved, one can inverse the problem and calculate the parameters which govern the process, that is: the degrees of melting corresponding to each lava, the initial source concentrations and partition coefficients for each element. This is a non-linear, strongly underdetermined problem, but when the set of trace elements is restricted to REE, taking advantage of the fact that their properties are smoothly related, the problem can be overconstrained by assuming a set of poorly constrained model parameters. Information contained in the data and in the external constraints can then be estimated and used to build a strategy of data acquisition. From a table of mineral-liquid partition coefficients, the mineralogies of the source and of the material entering the liquid can be calculated. The method has been tested on synthetic data sets representing natural cases, and proved to converge towards the real solution even when a very poor definition of the external constraints is introduced.The calculations have been applied to REE data on Grenada basanitoids (Shimizu and Arculus, 1976). It is confirmed that these lavas can be derived by 4 to 17% batch partial melting of a garnet lherzolite source (7% gt, 23% cpx) that is relatively enriched in light REE. This result is discussed in relation with Nd and Sr isotopic composition data.     

Comparison of structural and least-squares lines for estimating geologic relations

Two different goals in fitting straight lines to data are to estimate a true linear relation (physical law) and to predict values of the dependent variable with the smallest possible error. Regarding the first goal, a Monte Carlo study indicated that the structural-analysis (SA) method of fitting straight lines to data is superior to the ordinary least-squares (OLS) method for estimating true straight-line relations. Number of data points, slope and intercept of the true relation, and variances of the errors associated with the independent (X) and dependent (Y) variables influence the degree of agreement. For example, differences between the two line-fitting methods decrease as error in X becomes small relative to error in Y. Regarding the second goal—predicting the dependent variable—OLS is better than SA. Again, the difference diminishes as X takes on less error relative to Y. With respect to estimation of slope and intercept and prediction of Y, agreement between Monte Carlo results and large-sample theory was very good for sample sizes of 100, and fair to good for sample sizes of 20. The procedures and error measures are illustrated with two geologic examples.     

Garnet-cordierite-biotite equilibria in the Steinach aureole,Bavaria

Three garnet-biotite pairs and eleven garnet-cordierite-biotite triplets from the Steinach aureole (Oberpfalz, North-East Bavaria) were analyzed using an electron probe microanalyzer.The regional metamorphic muscovite-biotite schists contain garnets strongly zoned with Mn-Ca-rich centers and Fe-rich edges, the average composition being almandine 67 — spessartine 4 — pyrope 4 — grossular (+andradite) 25.The first contact garnet that is formed in mica schists of the outermost part of the aureole is small, virtually unzoned, and has an average composition of almandine 52 — spessartine 37 — pyrope 8 — grossular (+andradite) 3. With increasing metamorphic grade, there is a consistent trend to form garnets richer in Fe ending up with a composition almandine 84.5 — spessartine 5.5 — pyrope 7.5 — grossular (+andradite) 2.5. This trend is accompanied by a general increase in grain size and modal amount of garnet. Associated cordierites and biotites also become richer in Fe with increasing grade. While the garnets in the highest grade sillimanite hornfelses are poorly zoned, the transitional andalusite-sillimanite hornfelses contain garnets with distinct but variable zonation profiles.These facts can possibly be explained by the time-temperature relationships in the thermal aureole. In a phase diagram such as the Al-Fe-Mg-Mn tetrahedron, the limiting mineral compositions of a four-phase volume or a three-phase triangle are fixed by T and P (the latter remaining effectively constant within a thermal aureole). Thus, in garnet-cordierite-biotite assemblages, garnet zonation should be controlled by temperature variation rather than by a non-equilibrium depletion process. Taking into account the experimental data of Dahl (1968), a zoned garnet from a transitional andalusite-sillimanite hornfels would reflect a temperature increase of about 40° C during its growth. A hypothetical P-X diagram is proposed to show semi-quantitatively the compositional variation of garnet-cordierite pairs with varying pressures (T constant).     

U-Pb zircon,Rb-Sr and Sm-Nd geochronology of high- to very-high-pressure meta-acidic rocks from the western Alps

Three meta-acidic rocks from the western Italian Alps, a magnesiochloritoid-bearing metapelite from the Monte Rosa massif, a coesite-pyrope-quartzite from the Dora Maira massif and the Monte Mucrone granite in the Sesia Zone, have been studied by U-Pb zircon, Rb-Sr on whole-rock, apatite and phengite and Sm-Nd wholerock methods. The mineral parageneses of the investigated rocks indicate high- to very-high-pressure and medium-to-high-temperature metamorphism. This combined isotopic study has enabled us to constrain the ages of magmatic and metamorphic events and also to compare the behaviour of U-Pb zircon systems in three intensely metamorphosed areas of the Pennine domain. The U-Pb zircon data have yielded a magmatic age for the Monte Mucrone granite at 286±2 Ma. This result confirms the occurence of late-Hercynian magmatism in the Sesia Zone, as in other Austro-Alpine units and in other areas of the European crystalline basement. In the Monte Rosa massif, a geologically meaningless lower intercept age of 192±2 Ma has been interpreted as an artefact due to a complex evolution of the U-Pb zircon system. The magmatic shape of the zircons implies a magmatic or volcano-sedimentary protolith for this rock, originally considered as a metasediment. The very-high-pressure metamorphism in the Dora Maira quartzite has produced an opening of the U-Pb zircon system at 121+12–29 Ma. The Rb-Sr data support the occurence of high-grade metamorphism during Cretaceous times. Phengites model ages are slightly younger than the U-Pb zircon lower intercept ages due to cooling phenomena or possible response of the phengites to a later deformation. The Nd model ages from the whole-rock samples, as well as the U-Pb upper intercept ages from zircons of all three investigated rocks, indicate the presence of Proterozoic crustal components inherited from the precursors of these meta-acidic rocks. The studied zircon populations and their U-Pb systems apparently showed open-system behaviour only when affected by extreme metamorphic conditions (700–750° C, > 28 kbar), whereas eclogite-facies conditions of 500–550° C and 14–16 kbar were not enough to disturb significantly the U-Pb zircon evolution. It is also probable that the sedimentary or magmatic origin of the protoliths of these meta-acidic rocks, which involved different characteristics such as grain-size and fluid phase concentration and composition, could be another important factor controlling the U-Pb zircon system behaviour during metamorphic events.     

Capability of L-band SAR data in mapping of sedimentary formations of the Marmul region,Sultanate of Oman

This study demonstrates the capability of the longer wavelength (L-band) and fine mode images of the Phased Array type L-band Synthetic Aperture Radar (PALSAR), and the optical spectral bands of the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) to map sedimentary formations and structures of the Marmul region, Central Oman. Results of study shows that the principal components image of the visible and near infrared–shortwave infrared (VNIR-SWIR) spectral bands of the ASTER have potential to discriminate the occurrence and distribution of marine and Quaternary sedimentary formations of the region. The color composite image developed using PALSAR data (R:HH?+?HV; G:HH; B:HV) has capable to differentiate the occurrence of the sedimentary formations and structures of the region. We developed ratios images by summing the HH intensity as a numerator and the HV intensity as a denominator (R:HH+(HH?+?HV)/HV, G:HH?+?HV, B:HH/HV), and by summing the HV intensity as a numerator and the HH intensity as a denominator (R:HV+(HH?+?HV)/HH, G:HV?+?HH, B:HV/HH) to study better the sedimentary formations and structures. The images discriminated well the occurrence of the limestones and shale formations of the marine deposits, the wadi alluvium of the Quaternary deposits and the manmade building structures of the region. We studied the minerals of the sedimentary formations using VNIR-SWIR spectral bands of the ASTER by spectral angle mapper (SAM) image processing method. The study showed presence of carbonates, silicates and OH bearing aluminosilicate minerals in the formations. The image interpretations are verified in the field and confirmed by the laboratory studies.